Concurrence (quantum computing)

In quantum information science, the concurrence is a state invariant involving qubits.

Definition

the concurrence is an entanglement monotone defined for a mixed state of two qubits as:[1][2][3][4]

\mathcal{C}(\rho)\equiv\max(0,\lambda_1-\lambda_2-\lambda_3-\lambda_4)

in which \lambda_1,...,\lambda_4 are the eigenvalues, in decreasing order, of the Hermitian matrix

R = \sqrt{\sqrt{\rho}\tilde{\rho}\sqrt{\rho}}

with

\tilde{\rho} = (\sigma_{y}\otimes\sigma_{y})\rho^{*}(\sigma_{y}\otimes\sigma_{y})

the spin-flipped state of \rho, \sigma_y a Pauli spin matrix, and the eigenvalues listed in decreasing order.

Other formulations

Alternatively, the \lambda_{i}'s represent the square roots of the eigenvalues of the non-Hermitian matrix \rho\tilde{\rho}.[2] Note that each \lambda_{i} is a non-negative real number. From the concurrence, the entanglement of formation can be calculated.

Properties

For pure states, the concurrence is a polynomial SL(2,\mathbb{C})^{\otimes 2} invariant in the state's coefficients.[5] For mixed states, the concurrence can be defined by convex roof extension.[3]

For the concurrence, there is monogamy of entanglement,[6][7] that is, the concurrence of a qubit with the rest of the system cannot ever exceed the sum of the concurrences of qubit pairs which it is part of.

References

  1. ↑ Scott Hill and William K. Wootters, Entanglement of a Pair of Quantum Bits, 1997.
  2. 1 2 William K. Wootters, Entanglement of Formation of an Arbitrary State of Two Qubits 1998.
  3. 1 2 Roland Hildebrand, Concurrence revisited, 2007
  4. ↑ Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, Karol Horodecki, Quantum entanglement, 2009
  5. ↑ D. Ž. Ðoković and A. Osterloh, On polynomial invariants of several qubits, 2009
  6. ↑ Valerie Coffman, Joydip Kundu, and William K. Wootters, Distributed entanglement, 2000
  7. ↑ Tobias J. Osborne and Frank Verstraete, General Monogamy Inequality for Bipartite Qubit Entanglement, 2006
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